An alternative point of view in this context, is to consider instead probability measures on a space of suitable initial conditions; in the case of (REF ) this would be \(L^2_{\operatorname{div}}(D;U)\) . Equation (REF ) is then interpreted as a Liouville equation on an infinite dimensional function space and the solution is a mapping assigning to each time \(t\)  a probability measure on \(L^2_{\operatorname{div}}(D;U)\) . This setting was first considered by Prodi [1]} and later on extensively studied by Foiaş and collaborators [2]}, [3]}, [4]}, [5]}, see also [6]}. A closely related notion of statistical solutions was studied by Vishik and Fursikov [7]}. Foiaş and his collaborators proved existence of such solutions in 2D and 3D, uniqueness in 2D, and further properties related to turbulence [5]}. The relations between the Foiaş and Prodi notion of statistical solutions and the Vishik-Fursikov version were explored in [9]}, [10]}, [11]}. The latter work etends the notion of statistical solutions to other relevant PDEs in fluid mechanics.
